import sympy as sym
from IPython.display import display


# ###################################################################
def cubic_spline_clamped(
    xs: list[float], ys: list[float], B0: float, B1: float
) -> list[sym.Symbol]:
    """
    Cubic spline interpolation ``S``. Every two points are interpolated by a cubic polynomial
    ``S_j`` of the form ``S_j(x) = a_j + b_j(x - x_j) + c_j(x - x_j)^2 + d_j(x - x_j)^3.``

    xs must be different  but not necessarily ordered nor equally spaced.

    ## Parameters
    - xs, ys: points to be interpolated
    - B0, B1: derivatives at the first and last points

    ## Return
    - List of symbolic expressions for the cubic spline interpolation.
    """

    points = sorted(zip(xs, ys), key=lambda x: x[0])  # sort points by x
    xs = [x for x, _ in points]
    ys = [y for _, y in points]
    n = len(points) - 1  # number of splines
    h = [xs[i + 1] - xs[i] for i in range(n)]  # distances between  contiguous xs

    alpha = [0] * (n + 1)  # prealloc
    alpha[0] = 3 / h[0] * (ys[1] - ys[0]) - 3 * B0
    alpha[-1] = 3 * B1 - 3 / h[n - 1] * (ys[n] - ys[n - 1])

    for i in range(1, n):
        alpha[i] = 3 / h[i] * (ys[i + 1] - ys[i]) - 3 / h[i - 1] * (ys[i] - ys[i - 1])

    l = [2 * h[0]]
    u = [0.5]
    z = [alpha[0] / l[0]]

    for i in range(1, n):
        l += [2 * (xs[i + 1] - xs[i - 1]) - h[i - 1] * u[i - 1]]
        u += [h[i] / l[i]]
        z += [(alpha[i] - h[i - 1] * z[i - 1]) / l[i]]

    l.append(h[n - 1] * (2 - u[n - 1]))
    z.append((alpha[n] - h[n - 1] * z[n - 1]) / l[n])
    c = [0] * (n + 1)  # prealloc
    c[-1] = z[-1]

    x = sym.Symbol("x")
    splines = []
    for j in range(n - 1, -1, -1):
        c[j] = z[j] - u[j] * c[j + 1]
        b = (ys[j + 1] - ys[j]) / h[j] - h[j] * (c[j + 1] + 2 * c[j]) / 3
        d = (c[j + 1] - c[j]) / (3 * h[j])
        a = ys[j]
        print(j, a, b, c[j], d)
        S = a + b * (x - xs[j]) + c[j] * (x - xs[j]) ** 2 + d * (x - xs[j]) ** 3

        splines.append(S)
    splines.reverse()
    return splines

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from IPython.display import HTML
import sympy as sym

# Configuración para el primer conjunto de puntos (0,1), (1,5), (2,3)
xs_points = [0, 1, 2]
ys_points = [1, 5, 3]
B0 = 1  # Derivada fija en el primer punto
B1_values = np.linspace(-10, 10, 50)  # Rango de valores para B1

# Crear la figura
fig, ax = plt.subplots(figsize=(10, 6))
ax.set_xlim(-0.5, 2.5)
ax.set_ylim(-5, 10)
ax.grid(True)
ax.set_title('Splines cúbicos con tangentes en extremos')
line, = ax.plot([], [], 'b-', lw=2, label='Spline cúbico')
points = ax.plot(xs_points, ys_points, 'ro', ms=8, label='Puntos dados')

# Líneas tangentes
tangent0, = ax.plot([], [], 'g--', lw=1.5, label='Tangente en x=0')
tangent1, = ax.plot([], [], 'm--', lw=1.5, label=f'Tangente en x=2 (B1)')
ax.legend()

# Función para evaluar los splines
def evaluate_splines(splines, xs_points, x_range):
    y = []
    for x in x_range:
        for i in range(len(splines)):
            if x >= xs_points[i] and (i == len(splines)-1 or x <= xs_points[i+1]):
                y.append(float(splines[i].subs('x', x)))
                break
    return y

# Función para calcular la tangente en un punto
def get_tangent_line(x0, y0, slope, x_range, length=0.5):
    x_tangent = np.linspace(x0 - length, x0 + length, 2)
    y_tangent = y0 + slope * (x_tangent - x0)
    return x_tangent, y_tangent

# Función de inicialización
def init():
    line.set_data([], [])
    tangent0.set_data([], [])
    tangent1.set_data([], [])
    return line, tangent0, tangent1

# Función de animación
def animate(i):
    B1 = B1_values[i]
    
    # Calcular splines
    splines = cubic_spline_clamped(xs_points, ys_points, B0, B1)
    x_range = np.linspace(0, 2, 100)
    y_values = evaluate_splines(splines, xs_points, x_range)
    line.set_data(x_range, y_values)
    
    # Calcular y dibujar tangentes
    # Tangente en x=0 (usando B0)
    x_t0, y_t0 = get_tangent_line(xs_points[0], ys_points[0], B0, x_range)
    tangent0.set_data(x_t0, y_t0)
    
    # Tangente en x=2 (usando B1 actual)
    x_t1, y_t1 = get_tangent_line(xs_points[-1], ys_points[-1], B1, x_range)
    tangent1.set_data(x_t1, y_t1)
    
    ax.set_title(f'Splines cúbicos - B0={B0} (fijo), B1={B1:.2f}')
    return line, tangent0, tangent1

# Crear la animación
ani = FuncAnimation(fig, animate, frames=len(B1_values),
                    init_func=init, blit=True, interval=200)

# Mostrar la animación
HTML(ani.to_jshtml())

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# Configuración para el primer conjunto de puntos (0,1), (1,5), (2,3)
xs_points = [0, 1, 2]
ys_points = [-5, -4, 3]
B0 = 1  # Derivada fija en el primer punto
B1_values = np.linspace(-10, 10, 50)  # Rango de valores para B1

# Crear la figura
fig, ax = plt.subplots(figsize=(10, 6))
ax.set_xlim(-0.5, 2.5)
ax.set_ylim(-7, 10)
ax.grid(True)
ax.set_title('Splines cúbicos con tangentes en extremos')
line, = ax.plot([], [], 'b-', lw=2, label='Spline cúbico')
points = ax.plot(xs_points, ys_points, 'ro', ms=8, label='Puntos dados')

# Líneas tangentes
tangent0, = ax.plot([], [], 'g--', lw=1.5, label='Tangente en x=0')
tangent1, = ax.plot([], [], 'm--', lw=1.5, label=f'Tangente en x=2 (B1)')
ax.legend()

# Función para evaluar los splines
def evaluate_splines(splines, xs_points, x_range):
    y = []
    for x in x_range:
        for i in range(len(splines)):
            if x >= xs_points[i] and (i == len(splines)-1 or x <= xs_points[i+1]):
                y.append(float(splines[i].subs('x', x)))
                break
    return y

# Función para calcular la tangente en un punto
def get_tangent_line(x0, y0, slope, x_range, length=0.5):
    x_tangent = np.linspace(x0 - length, x0 + length, 2)
    y_tangent = y0 + slope * (x_tangent - x0)
    return x_tangent, y_tangent

# Función de inicialización
def init():
    line.set_data([], [])
    tangent0.set_data([], [])
    tangent1.set_data([], [])
    return line, tangent0, tangent1

# Función de animación
def animate(i):
    B1 = B1_values[i]
    
    # Calcular splines
    splines = cubic_spline_clamped(xs_points, ys_points, B0, B1)
    x_range = np.linspace(0, 2, 100)
    y_values = evaluate_splines(splines, xs_points, x_range)
    line.set_data(x_range, y_values)
    
    # Calcular y dibujar tangentes
    # Tangente en x=0 (usando B0)
    x_t0, y_t0 = get_tangent_line(xs_points[0], ys_points[0], B0, x_range)
    tangent0.set_data(x_t0, y_t0)
    
    # Tangente en x=2 (usando B1 actual)
    x_t1, y_t1 = get_tangent_line(xs_points[-1], ys_points[-1], B1, x_range)
    tangent1.set_data(x_t1, y_t1)
    
    ax.set_title(f'Splines cúbicos - B0={B0} (fijo), B1={B1:.2f}')
    return line, tangent0, tangent1

# Crear la animación
ani2 = FuncAnimation(fig, animate, frames=len(B1_values),
                    init_func=init, blit=True, interval=200)

# Mostrar la animación
HTML(ani2.to_jshtml())

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0 -5 1.0 -2.2500000000000004 2.2500000000000004

# Configuración para el primer conjunto de puntos 
xs_points = [0, 1, 2, 3]
ys_points = [-1, 1, 5, 2]
B0 = 1  # Derivada fija en el primer punto
B1_values = np.linspace(-10, 10, 50)  # Rango de valores para B1

# Crear la figura
fig, ax = plt.subplots(figsize=(10, 6))
ax.set_xlim(-0.5, 2.5)
ax.set_ylim(-5, 10)
ax.grid(True)
ax.set_title('Splines cúbicos con tangentes en extremos')
line, = ax.plot([], [], 'b-', lw=2, label='Spline cúbico')
points = ax.plot(xs_points, ys_points, 'ro', ms=8, label='Puntos dados')

# Líneas tangentes
tangent0, = ax.plot([], [], 'g--', lw=1.5, label='Tangente en x=0')
tangent1, = ax.plot([], [], 'm--', lw=1.5, label=f'Tangente en x=2 (B1)')
ax.legend()

# Función para evaluar los splines
def evaluate_splines(splines, xs_points, x_range):
    y = []
    for x in x_range:
        for i in range(len(splines)):
            if x >= xs_points[i] and (i == len(splines)-1 or x <= xs_points[i+1]):
                y.append(float(splines[i].subs('x', x)))
                break
    return y

# Función para calcular la tangente en un punto
def get_tangent_line(x0, y0, slope, x_range, length=0.5):
    x_tangent = np.linspace(x0 - length, x0 + length, 2)
    y_tangent = y0 + slope * (x_tangent - x0)
    return x_tangent, y_tangent

# Función de inicialización
def init():
    line.set_data([], [])
    tangent0.set_data([], [])
    tangent1.set_data([], [])
    return line, tangent0, tangent1

# Función de animación
def animate(i):
    B1 = B1_values[i]
    
    # Calcular splines
    splines = cubic_spline_clamped(xs_points, ys_points, B0, B1)
    x_range = np.linspace(0, 2, 100)
    y_values = evaluate_splines(splines, xs_points, x_range)
    line.set_data(x_range, y_values)
    
    # Calcular y dibujar tangentes
    # Tangente en x=0 (usando B0)
    x_t0, y_t0 = get_tangent_line(xs_points[0], ys_points[0], B0, x_range)
    tangent0.set_data(x_t0, y_t0)
    
    # Tangente en x=2 (usando B1 actual)
    x_t1, y_t1 = get_tangent_line(xs_points[-1], ys_points[-1], B1, x_range)
    tangent1.set_data(x_t1, y_t1)
    
    ax.set_title(f'Splines cúbicos - B0={B0} (fijo), B1={B1:.2f}')
    return line, tangent0, tangent1

# Crear la animación
ani3 = FuncAnimation(fig, animate, frames=len(B1_values),
                    init_func=init, blit=True, interval=200)

# Mostrar la animación
HTML(ani3.to_jshtml())

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Frontera condicionada¶